I wrote my Bachelor's thesis on the software that runs 5G networks.
4G (speaking about LTE) has been wildly successful since its introduction in 2009. It replaced the circuit switched network used in 2G and 3G with Evolved Packet Core, a network that delivered network packets directly to the phone using real IP addresses.
Now, a decade after 4G's release, technology has progressed a lot. What telecom companies call 5G are a collection of new technologies that are all coming out around the same time, but are mostly compatible with existing 4G phones and networks.
The big new technology is "5G new radio" (5G NR), which is a new optimized radio protocol that allows for faster speeds, lower power consumption, and the ability to use new extremely high bandwidth frequencies in the 50-400GHz range, on top of the existing 600-6000MHz range. Cellphones say that they're connected to 5G when they start speaking with the 5G NR protocol. I believe telecom companies can support 5G NR just by upgrading the cell tower antennas.
These newer antennas supporting this new radio protocol are also being built with beamforming tech, which allows towers to electronically "point" the antennas at nearby phones, improving signal.
Finally, the big upgrade underneath the surface is the migration to virtualize network functions, which is allowing telecom companies to run their networks using flexible lightweight computers inside the towers, instead of big bulky pre-built appliances in some big datacenter. This makes the network backbones faster, easier to program, and makes them cheaper for telecom companies to build out.
Wikipedia gives a bit of a summary about these features here: